Deny detective's private (6)
I believe the answer is:
disown
'deny' is the definition.
(thesaurus)
'detective's private' is the wordplay.
'detective's' becomes 'dis' (Detective Inspector).
'private' becomes 'own' (I've seen this before).
'dis'+'own'='DISOWN'
(Other definitions for disown that I've seen before include "Repudiate or refuse to acknowledge a relationship" , "Refuse to be connected with" , "Repudiate, cast off" , "Refuse responsibility for" , "Refuse to have anything to do with" .)